Free-Boundary Reformulation Methods

• Free-boundary reformulation is a strategy that converts moving interface problems with unknown boundaries into fixed-domain formulations using nonlocal PDEs, variational principles, and shape optimization.
• It enables rigorous analysis of complex phenomena like Hele-Shaw flows, Bernoulli problems, and debonding models while ensuring solution regularity and uniqueness.
• Advanced numerical techniques and variational approaches, including shape derivatives and SBV frameworks, offer robust and convergent schemes for simulating free-boundary dynamics.

A free-boundary reformulation is a mathematical paradigm in which problems originally posed with a moving or a priori unknown boundary (the "free boundary") are transformed into problems with a more amenable structure. This reformulation enables the problem to be studied in terms of alternative equations—typically nonlocal PDEs, variational principles, or shape optimization frameworks—while rigorously encoding the geometric evolution and solution regularity associated with the original free-boundary phenomena. Canonical cases include Hele-Shaw flows, Bernoulli-type free-boundary problems, and quasi-static debonding models, among others. The approach encompasses analytical reformulations (e.g., nonlocal fractional equations), variational minimization in nonsmooth function spaces (such as SBV), and advanced numerical methods exploiting shape derivatives and optimization strategies.

1. Free-Boundary Problems: Classical Formulation and Motivations

A free-boundary problem is characterized by the presence of an unknown spatial domain—typically the support of a solution or the interface between different phases—which evolves according to coupled PDEs and geometric (kinematic) laws. Representative examples are the Hele-Shaw flow for viscous fluid propagation, the Bernoulli problem for hydrodynamic jets and membranes, and thermal insulation models with energy-minimizing boundaries. The mathematical formulation often involves elliptic or parabolic PDEs in a time-dependent domain Ω(t), prescribed boundary conditions on fixed and free portions of ∂Ω(t), and a velocity law relating the boundary's normal movement to solution gradients (e.g., $V_n = \partial_n p)$. Analytical challenges manifest in nonlinearity, nonlocality, and regularity concerns of the interface, as well as difficulties in establishing uniqueness and stability.

2. Nonlocal Parabolic Reformulation for Graph-Like Free Boundaries

Chang-Lara, Guillen, and Schwab introduced a powerful methodology for recasting one- and two-phase free-boundary flows, especially in settings where the free boundary remains a global graph over $\mathbb{R}^d$ (Chang-Lara et al., 2018). The procedure is as follows:

• For the one-phase case (Hele-Shaw flows), consider solutions $U(X,t)$ in domains such as $\mathbb{R}^d \times [0,\infty)$, with prescribed Dirichlet conditions at fixed boundary and a moving Neumann condition at the free boundary.
• Given a graph parameterization $x_{d+1} = f(x,t)$, one defines the bulk domain $D_f$ and solves the associated elliptic PDE with boundary data $U_f(x,0) = 1$, $U_f(x, f(x)) = 0$.
• The free-boundary velocity is given by a nonlocal operator $I(f)(x) = \partial_n U_f(x, f(x))$ and the evolution law $$\partial_t f(x,t) = I(f(·,t))(x)\sqrt{1+|\nabla f|^2}$$.
• For two-phase analogs, replacements of the elliptic operators and coupling function $G$ allow for further generalization.

Key properties:

• The nonlocal Dirichlet-to-Neumann map $I$ possesses the Global Comparison Property (GCP): if $u \leq v$ everywhere and $u(x_0)=v(x_0)$, then $I(u)(x_0) \leq I(v)(x_0)$.
• The operator $I$ is Lipschitz on suitable function classes and admits a Courrége-type min–max representation involving Lévy measures, leading to a fully nonlinear, integro-differential parabolic PDE for $f(x,t)$.
• Viscosity solutions to the reformulated equation are equivalent to viscosity solutions of the original geometric free-boundary problem, guaranteeing uniqueness via comparison principles and propagation of regularity.
• Special cases include linear Hele-Shaw flow, for which $I(f)$ reduces to the half-Laplacian in the small-slope limit.

This approach unifies a wide array of PDE-driven interface evolution problems, provides new existence and regularity results, and facilitates analysis via the theory of nonlocal parabolic equations.

3. Variational Reformulation and SBV Methods

Several free-boundary problems, especially those with energy minimization as a driving mechanism, admit reformulation as variational problems in SBV (Special Functions of Bounded Variation) or related nonsmooth function spaces (Acampora et al., 2022). Paradigmatic examples include thermal insulation and Bernoulli-type problems:

• Classical formulations seek an optimal domain $A$ and function $u$ minimizing bulk and surface energy terms, often under nonlinear PDE and boundary constraints.
• The SBV reformulation encodes the unknown domain $A$ and its free boundary $\partial A$ as the positivity set and jump set $J_v$ of a function $v \in SBV(\mathbb{R}^n)$, leading to a single minimization problem:

$$\mathcal{F}(v) = \int_{\mathbb{R}^n} |\nabla v|^p dx + \beta \int_{J_v} (\underline{v}^q + \overline{v}^q) d\mathcal{H}^{n-1} + |{v>0} \setminus \Omega|$$

with constraints $v=1$ on $\Omega$, $0 \leq v \leq 1$ a.e.

• Existence theorems establish the existence of minimizers under appropriate exponent conditions $(p,q)$, nondegeneracy (uniform positivity on the support), and uniform density estimates for the jump set (free boundary), ensuring sharp measure-theoretic control of the boundary's regularity.

This variational approach simplifies existence proofs and enables robust analysis via compactness and $\Gamma$-convergence arguments.

4. Shape Optimization and Numerical Schemes

Modern free-boundary reformulation often incorporates shape optimization frameworks, allowing for systematic numerical approximation and iterative boundary evolution. Techniques such as the CutFEM and shape-Newton methods have demonstrated flexibility and conditioning benefits (Burman et al., 2016, Fan et al., 2023):

• Shape derivatives and Hadamard formulas provide first-order (and second-order in Newton-type schemes) sensitivity of cost functionals or PDE weak forms with respect to domain perturbations, enabling rigorous gradient-based and Newton-based boundary updates.
• Riesz representation in Hilbert spaces (e.g., $H^1$) is used to generate smooth velocity fields for level-set or mesh node propagation.
• Stabilized finite element discretizations, such as Nitsche-type penalty terms for cut elements in CutFEM, guarantee robustness when the boundary slices through the background mesh arbitrarily and maintain optimal order conditioning.
• Iterative schemes converge rapidly; e.g., shape-Newton methods exhibit superlinear convergence to the free-boundary configuration, and CutFEM approaches achieve $O(h^{1/2})$ convergence in the boundary update step.

These frameworks are employed for time-dependent interface propagation, steady Bernoulli problems, and multi-physics settings such as plasma equilibria in Grad–Shafranov solvers (Serino et al., 3 Jul 2024).

5. Weak Formulations and Alternative Dynamics

Free-boundary reformulation may also employ weak PDE frameworks where the free boundary is encoded not as an explicit interface but via complementarity conditions and measures acting only on the support's edge (Atar, 2020):

• For selection-dominated particle systems, the PDE for density couples to a nonnegative measure encoding removal or mass exchange,

$$\partial_t u - \frac{1}{2}\partial_x^2 u - b(x)\partial_x u - \frac{1}{2}c'(x) u = \alpha - \beta$$

subject to the constraint $\beta(\{(x,t): U(x,t)>0\})=0$, i.e., measure is active only at the interface.

• When a classical interface exists, the measure becomes singularly supported at the boundary, recovering standard Stefan or Hele-Shaw laws.
• Robust uniqueness is ensured by mass transport barrier arguments, and hydrodynamic limits from microscopic stochastic models are characterized by convergence to this weak formulation.

Such approaches generalize well to non-smooth, possibly measure-valued, or nonclassical interface dynamics.

6. Applications and Extensions

Free-boundary reformulation has advanced theory and practice in numerous domains:

• Quasistatic debonding and rate-independent evolutions are reduced to sequential Bernoulli-type free-boundary energy minimizations, facilitating long-time existence and continuous trajectory regularity (Maggiorelli et al., 21 Mar 2025).
• Boundary layer theory (e.g., extended Blasius problems) benefits from reformulations where the far-field boundary condition is replaced by conditions at a numerically determined free-boundary, enhancing the accuracy and tractability of the solution via, e.g., collocation methods (Fazio, 2020).
• Non-Newtonian and plasticity-driven flows (Drucker–Prager) with variational free-boundary reformulation yield compact support and sharp estimates for critical angles of flow initiation (Ntovoris et al., 2016).
• Numerical simulation of strongly nonlinear or degenerate diffusion phenomena (e.g., tumor growth with $m\gg 1$ in porous medium models) is enabled by prediction–correction schemes that naturally degenerate to sharp interface Hele-Shaw evolution in the limit (Liu et al., 2017).
• Boundary value problems on semi-infinite intervals are reformulated by replacing conditions at infinity with free-boundary conditions for an unknown finite truncation, as in the convergence theorem of Fazio (Fazio, 2020, Fazio, 2014).

Pervasive across applications are common principles: the introduction of a small parameter for "tolerance" at the free boundary, the use of group-invariance non-iterative transformation methods (especially when Lie symmetries exist), and the exploitation of variational or weak forms for both analytical and computational study.

7. Analytical and Computational Impact

Free-boundary reformulation has provided a unified and rigorous platform for analytical progress and computational efficiency in interface problems:

• It facilitates a reduction from high-dimensional geometrically evolving domains to lower-dimensional parabolic or integro-differential equations with proven regularity and comparison principles (Chang-Lara et al., 2018).
• Variational and SBV formulations yield strong existence and measure-theoretic regularity results, including uniform geometric density estimates for the free boundary (Acampora et al., 2022).
• Advanced numerical methods are enabled: robust mesh-independent solvers, scalable preconditioned Newton strategies for nonlinear plasma equilibria, and shape optimization with high-order convergence (Serino et al., 3 Jul 2024, Burman et al., 2016, Rabago et al., 2023, Fan et al., 2023).
• Weak measure-driven approaches provide model flexibility and capture non-classical interface movements in stochastic or branching systems (Atar, 2020).

The reformulation paradigm substantially broadens the scope of tractable free-boundary phenomena and underpins ongoing research into higher-order regularity, nonlocal dynamics, and multi-physics coupling.